Discussion
Consider the following two statements :
(1) The correlation coefficient between two variables X and Y is the geometric mean of two regression coefficients βYX and βXY.
(2) The arithmetic mean of the two regression coefficients βYX and βXY is greater than or equal to the correlation coefficient between X and Y.
Which of the above statement(s) is/are true ?
Answer: C
The key gives both (1) and (2).
(1) The regression coefficients are \(\beta_{YX} = r\frac{\sigma_{Y}}{\sigma_{X}}\) and \(\beta_{XY} = r\frac{\sigma_{X}}{\sigma_{Y}}\), so \(\beta_{YX}\beta_{XY} = r^{2}\). The correlation coefficient is their geometric mean, \(r = \pm\sqrt{\beta_{YX}\beta_{XY}}\), taking the common sign of the two coefficients.
(2) is the standard textbook property that follows from the AM-GM inequality: for positive coefficients,
\[ \begin{aligned} \frac{\beta_{YX} + \beta_{XY}}{2} &\ge \sqrt{\beta_{YX}\beta_{XY}} \\ &= r \end{aligned} \]
Textbooks state it for the usual case of positive correlation; for negative correlation it holds for the magnitudes. On that reading both statements are true, so the options naming only one, or neither, are wrong.
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